# Substructural logic

In mathematical logic, in particular in connection with proof theory, a number of substructural logics have been introduced, as systems of propositional calculus that are weaker than the conventional one. They differ in having fewer structural rules available: the concept of structural rule is based on the sequent presentation, rather than the natural deduction formulation. Two of the more significant substructural logics are relevant logic and linear logic.

In a sequent calculus, one writes each line of a proof as

$\displaystyle \Gamma\vdash\Sigma$ .

Here the structural rules are rules for rewriting the LHS Γ of the sequent, initially conceived of as a string of propositions. The standard interpretation of this string is as conjunction: we expect to read

$\displaystyle \mathcal A,\mathcal B \vdash\mathcal C$

as the sequent notation for

(A and B) implies C.

Here we are taking the RHS Σ to be a single proposition C (which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol.

Since conjunction is a commutative and associative operation, the formal setting-up of sequent theory normally includes structural rules for rewriting the sequent Γ accordingly - for example for deducing

$\displaystyle \mathcal B,\mathcal A\vdash\mathcal C$

from

$\displaystyle \mathcal A,\mathcal B\vdash\mathcal C$ .

There are further structural rules corresponding to the idempotent and monotonic properties of conjunction: from

$\displaystyle \Gamma,\mathcal A,\mathcal A,\Delta\vdash\mathcal C$

we can deduce

$\displaystyle \Gamma,\mathcal A,\Delta\vdash\mathcal C$ .

Also from

$\displaystyle \Gamma,\mathcal A,\Delta\vdash\mathcal C$

one can deduce, for any B,

$\displaystyle \Gamma,\mathcal A,\mathcal B,\Delta\vdash\mathcal C$ .

Linear logic, in which duplicated hypotheses 'count' differently from single occurrences, leaves out both of these rules, while relevant (or relevance) logics merely leaves out the latter rule, on the ground that B is clearly irrelevant to the conclusion.

These are basic examples of structural rules. It is not that these rules are contentious, when applied in conventional propositional calculus. They occur naturally in proof theory, and were first noticed there (before receiving a name).